On a Two-Component Shallow-Water Model with the Weak Coriolis and Equatorial Undercurrent Effects

Abstract

The present paper studies a two-component mathematical model representing shallow-water wave propagation primarily in equatorial ocean regions, incorporating the effects of weak Coriolis force and equatorial undercurrent. We start with the Green–Naghdi type equations under the weak Coriolis and equatorial undercurrent effects from the Euler equations, then the two-component Camassa–Holm system with the two effects is derived by truncating asymptotic expansions of the quantities to the appropriate order. Analytically, we study the mathematical properties of the solutions to the two-component Camassa–Holm system including the ill-posedness of the solutions in Besov spaces \(B^{s}_{p,\infty }\times B^{s-1}_{p,\infty }\) with \(1\le p\le \infty \) and \(s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} \), the Hölder continuity of the data-to-solution map in Besov spaces \(B^{s}_{p,r}\times B^{s-1}_{p,r}\) with \(1\le p,r\le \infty \) and \(s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} \). We then investigate the Gevrey regularity and analyticity of the system in \({G_{\delta ,s}^{\gamma }}\times {G_{\delta ,s-1}^{\gamma }}\) with \(\delta \ge 1,\ \nu>\gamma >0\) and \(s>\frac{5}{2}\). Finally, we provide the persistence properties and the spatial asymptotic profiles of the solutions in weighted spaces \(L ^ p_{\phi }=L^p(\mathbb {R},\phi ^pdx)\).